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Mysterious Numbers

Take any 3 digit number whose digits are not all the same. Numbers like 027, 889, and 534 are all valid examples. Now let us just start with 027. We are going to play a game. First rewrite 027 by ordering the numbers in descending order, 720. Then subtract from it the number made by ordering them in ascending order, 027. We get 720-027 = 693. We will continue this until we get a repeating number.

027 →693→594→495→495

Okay, so we got 495. Let’s try again with 889

889→099→891→792→693→594→495→495

Again! 495 shows up. So let us try with 534

534→198→792→693→594→495→495

And yet again our game terminates with 495. Believe it or not, this happens with every 3-digit number whose digits are not the same. Since there are only 990 possible starting points, we can actually prove this by exhaustion. But we definitely shouldn’t do it by hand, unless boredom should overtake us. But 495 isn’t the only mysterious fixed number. 6174 also fits this bill. And it is also provable by computer in a short amount of time. Numbers are a fascinating part of mathematics and have many interesting properties. They are surprising us everyday! Please check out http://oeis.org/A099009 for more of these numbers and an explanation of why.